DOCSLIB.ORG

Eienstein Field Equations and Heisenberg's Principle Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Eienstein Field Equations and Heisenberg's Principle Of International Journal of Scientific and Research Publications, Volume 2, Issue 9, September 2012 1 ISSN 2250-3153 EIENSTEIN FIELD EQUATIONS AND HEISENBERG’S PRINCIPLE OF UNCERTAINLY THE CONSUMMATION OF GTR AND UNCERTAINTY PRINCIPLE 1DR K N PRASANNA KUMAR, 2PROF B S KIRANAGI AND 3PROF C S BAGEWADI ABSTRACT: The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction (e&eb) of gravitation as a result of space time being curved by matter and energy. First published by Einstein in 1915 as a tensor equation, the EFE equate spacetime curvature (expressed by the Einstein tensor) with (=) the energy and momentum tensor within that spacetime (expressed by the stress–energy tensor).Both space time curvature tensor and energy and momentum tensor is classified in to various groups based on which objects they are attributed to. It is to be noted that the total amount of energy and mass in the Universe is zero. But as is said in different context, it is like the Bank Credits and Debits, with the individual debits and Credits being conserved, holistically, the conservation and preservation of Debits and Credits occur, and manifest in the form of General Ledger. Transformations of energy also take place individually in the same form and if all such transformations are classified and written as a Transfer Scroll, it should tally with the total, universalistic transformation. This is a very important factor to be borne in mind. Like accounts are classifiable based on rate of interest, balance standing or the age, we can classify the factors and parameters in the Universe, be it age, interaction ability, mass, energy content. Even virtual particles could be classified based on the effects it produces. These aspects are of paramount importance in the study. When we write A+b+5, it means that we are adding A to B or B to A until we reach 5. Similarly, if we write A-B=0, it means we are taking away B from A and there may be time lag until we reach zero. There may also be cases in which instantaneous results are reached, which however do not affect the classification. By means of such a classification we obtain the values of Einstein Tensor and Momentum Energy Tensor, which are in fact the solutions to the Einstein‘s Field Equation. Terms ―e‖ and ―eb‖ are used for better comprehension of the lay reader. It has no other attribution or ascription whatsoever in the context of the paper. For the sake of simplicity, we shall take the equality case of Heisenberg’s Principle Of Uncertainty for easy consolidation and consubstantiation process. The “greater than” case can be attended to in a similar manner, with the symbolof”greater than” incorporated in the paper series. INTRODUCTION: Similar to the way that electromagnetic fields are determined (eb) using charges and currents via Maxwell's equations, the EFE are used to determine the spacetime geometry resulting from the presence of mass-energy and linear momentum, that is, they (eb) determine the metric of spacetime for a given arrangement of stress–energy in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation. As well as obeying local energy-momentum conservation, the EFE reduce to Newton's www.ijsrp.org International Journal of Scientific and Research Publications, Volume 2, Issue 9, September 2012 2 ISSN 2250-3153 law of gravitation where the gravitational field is weak and velocities are much less than the speed of light. Solution techniques for the EFE include simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied as they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the actual spacetime as flat spacetime with a small deviation, leading to the linearised EFE. These equations are used to study phenomena such as gravitational waves. Mathematical form The Einstein field equations (EFE) may be written in the form: where is the Ricci curvature tensor, the scalar curvature, the metric tensor, is the cosmological constant, is Newton's gravitational constant, the speed of light,in vacuum, and the stress–energy tensor. The EFE is a tensor equation relating a set of symmetric 4 x 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge fixing degrees of freedom, which correspond to the freedom to choose a coordinate system. Although the Einstein field equations were initially formulated in the context of a four- dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T is identically zero) define Einstein manifolds. Despite the simple appearance of the equations they are, in fact, quite complicated. Given a specified distribution of(e&eb) matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor , as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of 10 coupled, nonlinear, hyperbolic-elliptic partial differential equations. One can write the EFE in a more compact form by defining the Einstein tensor Which is a symmetric second-rank tensor that is a function of the metric? The EFE can then be written as Using geometrized units where G = c = 1, this can be rewritten as The expression on the left represents the curvature of spacetime as (eb) determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines (eb) the curvature of spacetime.Or, curvature of space and time dictates the diffusion of matter energy. These equations, together with the geodesic equation, which dictates how freely-falling moves through space-time matter, form the core of the mathematical formulation of general relativity. www.ijsrp.org International Journal of Scientific and Research Publications, Volume 2, Issue 9, September 2012 3 ISSN 2250-3153 Sign convention The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, and S3): The third sign above is related to the choice of convention for the Ricci tensor: With these definitions Misner, Thorne, and Wheeler classify themselves as , whereas Weinberg (1972) is , Peebles (1980) and Efstathiou (1990) are while Peacock (1994), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) are . Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative The sign of the (very small) cosmological term would change in both these versions, if the +−−− metric sign convention is used rather than the MTW −+++ metric sign convention adopted here. Equivalent formulations Taking the trace of both sides of the EFE one gets Which simplifies to If one adds times this to the EFE, one gets the following equivalent "trace-reversed" form Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace in the expression on the right with the Minkowski metric without significant loss of accuracy). www.ijsrp.org International Journal of Scientific and Research Publications, Volume 2, Issue 9, September 2012 4 ISSN 2250-3153 The cosmological constant Einstein modified his original field equations to include a cosmological term proportional to the metric It is to be noted that even constants like gravitational field, cosmological constant, depend upon the objects for which they are taken in to consideration and total of these can be classified based on the parameterization of objects. The constant is the cosmological constant. Since is constant, the energy conservation law is unaffected. The cosmological constant term was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations of distant galaxies by Hubble a decade later confirmed that our universe is, in fact, not static but expanding. So was abandoned, with Einstein calling it the "biggest blunder [he] ever made". For many years the cosmological constant was almost universally considered to be 0.Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. Indeed, recent improved astronomical techniques have found that a positive value of is needed to explain the accelerating universe Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side, written as part of the stress–energy tensor: The resulting vacuum energy is constant and given by The existence of a cosmological constant is thus equivalent to the existence of a non-zero vacuum energy.
Load more

Recommended publications
  • General Relativity
    GENERAL RELATIVITY IAN LIM LAST UPDATED JANUARY 25, 2019 These notes were taken for the General Relativity course taught by Malcolm Perry at the University of Cambridge as part of the Mathematical Tripos Part III in Michaelmas Term 2018. I live-TEXed them using TeXworks, and as such there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Many thanks to Arun Debray for the LATEX template for these lecture notes: as of the time of writing, you can find him at https://web.ma.utexas.edu/users/a.debray/. Contents 1. Friday, October 5, 2018 1 2. Monday, October 8, 2018 4 3. Wednesday, October 10, 20186 4. Friday, October 12, 2018 10 5. Wednesday, October 17, 2018 12 6. Friday, October 19, 2018 15 7. Monday, October 22, 2018 17 8. Wednesday, October 24, 2018 22 9. Friday, October 26, 2018 26 10. Monday, October 29, 2018 28 11. Wednesday, October 31, 2018 32 12. Friday, November 2, 2018 34 13. Monday, November 5, 2018 38 14. Wednesday, November 7, 2018 40 15. Friday, November 9, 2018 44 16. Monday, November 12, 2018 48 17. Wednesday, November 14, 2018 52 18. Friday, November 16, 2018 55 19. Monday, November 19, 2018 57 20. Wednesday, November 21, 2018 62 21. Friday, November 23, 2018 64 22. Monday, November 26, 2018 67 23. Wednesday, November 28, 2018 70 Lecture 1. Friday, October 5, 2018 Unlike in previous years, this course is intended to be a stand-alone course on general relativity, building up the mathematical formalism needed to construct the full theory and explore some examples of interesting spacetime metrics.
    [Show full text]
  • 1 Electromagnetic Fields and Charges In
    Electromagnetic fields and charges in ‘3+1’ spacetime derived from symmetry in ‘3+3’ spacetime J.E.Carroll Dept. of Engineering, University of Cambridge, CB2 1PZ E-mail: [email protected] PACS 11.10.Kk Field theories in dimensions other than 4 03.50.De Classical electromagnetism, Maxwell equations 41.20Jb Electromagnetic wave propagation 14.80.Hv Magnetic monopoles 14.70.Bh Photons 11.30.Rd Chiral Symmetries Abstract: Maxwell’s classic equations with fields, potentials, positive and negative charges in ‘3+1’ spacetime are derived solely from the symmetry that is required by the Geometric Algebra of a ‘3+3’ spacetime. The observed direction of time is augmented by two additional orthogonal temporal dimensions forming a complex plane where i is the bi-vector determining this plane. Variations in the ‘transverse’ time determine a zero rest mass. Real fields and sources arise from averaging over a contour in transverse time. Positive and negative charges are linked with poles and zeros generating traditional plane waves with ‘right-handed’ E, B, with a direction of propagation k. Conjugation and space-reversal give ‘left-handed’ plane waves without any net change in the chirality of ‘3+3’ spacetime. Resonant wave-packets of left- and right- handed waves can be formed with eigen frequencies equal to the characteristic frequencies (N+½)fO that are observed for photons. 1 1. Introduction There are many and varied starting points for derivations of Maxwell’s equations. For example Kobe uses the gauge invariance of the Schrödinger equation [1] while Feynman, as reported by Dyson, starts with Newton’s classical laws along with quantum non- commutation rules of position and momentum [2].
    [Show full text]
  • Math 865, Topics in Riemannian Geometry
    Math 865, Topics in Riemannian Geometry Jeff A. Viaclovsky Fall 2007 Contents 1 Introduction 3 2 Lecture 1: September 4, 2007 4 2.1 Metrics, vectors, and one-forms . 4 2.2 The musical isomorphisms . 4 2.3 Inner product on tensor bundles . 5 2.4 Connections on vector bundles . 6 2.5 Covariant derivatives of tensor fields . 7 2.6 Gradient and Hessian . 9 3 Lecture 2: September 6, 2007 9 3.1 Curvature in vector bundles . 9 3.2 Curvature in the tangent bundle . 10 3.3 Sectional curvature, Ricci tensor, and scalar curvature . 13 4 Lecture 3: September 11, 2007 14 4.1 Differential Bianchi Identity . 14 4.2 Algebraic study of the curvature tensor . 15 5 Lecture 4: September 13, 2007 19 5.1 Orthogonal decomposition of the curvature tensor . 19 5.2 The curvature operator . 20 5.3 Curvature in dimension three . 21 6 Lecture 5: September 18, 2007 22 6.1 Covariant derivatives redux . 22 6.2 Commuting covariant derivatives . 24 6.3 Rough Laplacian and gradient . 25 7 Lecture 6: September 20, 2007 26 7.1 Commuting Laplacian and Hessian . 26 7.2 An application to PDE . 28 1 8 Lecture 7: Tuesday, September 25. 29 8.1 Integration and adjoints . 29 9 Lecture 8: September 23, 2007 34 9.1 Bochner and Weitzenb¨ock formulas . 34 10 Lecture 9: October 2, 2007 38 10.1 Manifolds with positive curvature operator . 38 11 Lecture 10: October 4, 2007 41 11.1 Killing vector fields . 41 11.2 Isometries . 44 12 Lecture 11: October 9, 2007 45 12.1 Linearization of Ricci tensor .
    [Show full text]
  • 3+1 Formalism and Bases of Numerical Relativity
    3+1 Formalism and Bases of Numerical Relativity Lecture notes Eric´ Gourgoulhon Laboratoire Univers et Th´eories, UMR 8102 du C.N.R.S., Observatoire de Paris, Universit´eParis 7 arXiv:gr-qc/0703035v1 6 Mar 2007 F-92195 Meudon Cedex, France [email protected] 6 March 2007 2 Contents 1 Introduction 11 2 Geometry of hypersurfaces 15 2.1 Introduction.................................... 15 2.2 Frameworkandnotations . .... 15 2.2.1 Spacetimeandtensorfields . 15 2.2.2 Scalar products and metric duality . ...... 16 2.2.3 Curvaturetensor ............................... 18 2.3 Hypersurfaceembeddedinspacetime . ........ 19 2.3.1 Definition .................................... 19 2.3.2 Normalvector ................................. 21 2.3.3 Intrinsiccurvature . 22 2.3.4 Extrinsiccurvature. 23 2.3.5 Examples: surfaces embedded in the Euclidean space R3 .......... 24 2.4 Spacelikehypersurface . ...... 28 2.4.1 Theorthogonalprojector . 29 2.4.2 Relation between K and n ......................... 31 ∇ 2.4.3 Links between the and D connections. .. .. .. .. .. 32 ∇ 2.5 Gauss-Codazzirelations . ...... 34 2.5.1 Gaussrelation ................................. 34 2.5.2 Codazzirelation ............................... 36 3 Geometry of foliations 39 3.1 Introduction.................................... 39 3.2 Globally hyperbolic spacetimes and foliations . ............. 39 3.2.1 Globally hyperbolic spacetimes . ...... 39 3.2.2 Definition of a foliation . 40 3.3 Foliationkinematics .. .. .. .. .. .. .. .. ..... 41 3.3.1 Lapsefunction ................................. 41 3.3.2 Normal evolution vector . 42 3.3.3 Eulerianobservers ............................. 42 3.3.4 Gradients of n and m ............................. 44 3.3.5 Evolution of the 3-metric . 45 4 CONTENTS 3.3.6 Evolution of the orthogonal projector . ....... 46 3.4 Last part of the 3+1 decomposition of the Riemann tensor .
    [Show full text]
  • Chapter 13 Curvature in Riemannian Manifolds
    Chapter 13 Curvature in Riemannian Manifolds 13.1 The Curvature Tensor If (M, , )isaRiemannianmanifoldand is a connection on M (that is, a connection on TM−), we− saw in Section 11.2 (Proposition 11.8)∇ that the curvature induced by is given by ∇ R(X, Y )= , ∇X ◦∇Y −∇Y ◦∇X −∇[X,Y ] for all X, Y X(M), with R(X, Y ) Γ( om(TM,TM)) = Hom (Γ(TM), Γ(TM)). ∈ ∈ H ∼ C∞(M) Since sections of the tangent bundle are vector fields (Γ(TM)=X(M)), R defines a map R: X(M) X(M) X(M) X(M), × × −→ and, as we observed just after stating Proposition 11.8, R(X, Y )Z is C∞(M)-linear in X, Y, Z and skew-symmetric in X and Y .ItfollowsthatR defines a (1, 3)-tensor, also denoted R, with R : T M T M T M T M. p p × p × p −→ p Experience shows that it is useful to consider the (0, 4)-tensor, also denoted R,givenby R (x, y, z, w)= R (x, y)z,w p p p as well as the expression R(x, y, y, x), which, for an orthonormal pair, of vectors (x, y), is known as the sectional curvature, K(x, y). This last expression brings up a dilemma regarding the choice for the sign of R. With our present choice, the sectional curvature, K(x, y), is given by K(x, y)=R(x, y, y, x)but many authors define K as K(x, y)=R(x, y, x, y). Since R(x, y)isskew-symmetricinx, y, the latter choice corresponds to using R(x, y)insteadofR(x, y), that is, to define R(X, Y ) by − R(X, Y )= + .
    [Show full text]
  • Linear Degeneracy in Multidimensions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Loughborough University Institutional Repository Linear degeneracy in multidimensions by Jonathan Moss A Doctoral Thesis Submitted in partial fulfillment of the requirements for the award of Doctor of Philosophy of Loughborough University September 2015 c J J Moss 2015 Abstract Linear degeneracy of a PDE is a concept that is related to a number of interesting geometric constructions. We first take a quadratic line complex, which is a three- parameter family of lines in projective space P3 specified by a single quadratic relation in the Pl¨ucker coordinates. This complex supplies us with a conformal structure in P3. With this conformal structure, we associate a three-dimensional second order quasilin- ear wave equation. We show that any PDE arising in this way is linearly degenerate, furthermore, any linearly degenerate PDE can be obtained by this construction. We classify Segre types of quadratic complexes for which the structure is conformally flat, as well as Segre types for which the corresponding PDE is integrable. These results were published in [1]. We then introduce the notion of characteristic integrals, discuss characteristic integrals in 3D and show that, for certain classes of second-order linearly degenerate dispersionless integrable PDEs, the corresponding characteristic integrals are parameterised by points on the Veronese variety. These results were published in [2]. Keywords Second order PDEs, hydrodynamic reductions, integrability, conformal structures, quadratic line complexes, linear degeneracy, characteristic integrals, principal symbol. 1 Acknowledgments I would like to express many thanks to Prof E.V.
    [Show full text]
  • Einstein Manifolds As Yang-Mills Instantons
    arXiv:1101.5185 Einstein Manifolds As Yang-Mills Instantons a b John J. Oh ∗ and Hyun Seok Yang † a Division of Computational Sciences in Mathematics, National Institute for Mathematical Sciences, Daejeon 305-340, Korea b Institute for the Early Universe, Ewha Womans University, Seoul 120-750, Korea b Center for Quantum Spacetime, Sogang University, Seoul 121-741, Korea ABSTRACT It is well-known that Einstein gravity can be formulated as a gauge theory of Lorentz group where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths. One can then pose an interesting question: What is the Einstein equation from the gauge theory point of view? Or equivalently, what is the gauge theory object corresponding to Einstein manifolds? We show that the Einstein equations in four dimensions are precisely self-duality equa- tions in Yang-Mills gauge theory and so Einstein manifolds correspond to Yang-Mills instantons in SO(4) = SU(2) SU(2) gauge theory. Specifically, we prove that any Einstein manifold L × R with or without a cosmological constant always arises as the sum of SU(2)L instantons and SU(2)R anti-instantons. This result explains why an Einstein manifold must be stable because two kinds of arXiv:1101.5185v4 [hep-th] 2 Jul 2013 instantons belong to different gauge groups, instantons in SU(2)L and anti-instantons in SU(2)R, and so they cannot decay into a vacuum. We further illuminate the stability of Einstein manifolds by showing that they carry nontrivial topological invariants. Keywords: Einstein manifold, Yang-Mills instanton, Self-duality May 30, 2018 ∗[email protected] †[email protected] 1 Introduction It seems that the essence of the method of physics is inseparably connected with the problem of interplay between local and global aspects of the world’s structure, as saliently exemplified in the index theorem of Dirac operators.
    [Show full text]
  • Arxiv:1408.0902V1 [Math.DG] 5 Aug 2014 Bandb Usy[ Gursky by Obtained Ufcsaeawy Ofral A,Hnei Sntrlt L to Natural Is It [ Hence Kuiper flat, Conformally Case
    ON CONFORMALLY FLAT MANIFOLDS WITH CONSTANT POSITIVE SCALAR CURVATURE GIOVANNI CATINO Abstract. We classify compact conformally flat n-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are − covered isometrically by either Sn with the round metric, S1 × Sn 1 with the product metric − or S1 × Sn 1 with a rotationally symmetric Derdzi´nski metric. Key Words: conformally flat manifold, rigidity AMS subject classification: 53C20, 53C21 1. Introduction In this paper, we study compact conformally flat Riemannian manifolds, i.e. compact man- ifolds whose metrics are locally conformally equivalent to the Euclidean metric. Riemannian surfaces are always conformally flat, hence it is natural to look to the higher-dimensional case. Kuiper [21] was the first who studied global properties of this class of manifolds. He showed that every compact, simply connected, conformally flat manifolds is conformally dif- feomorphic to the round sphere Sn. In the last years, much attention has been given to the classification of conformally flat manifolds under topological and/or geometrical assumptions. From the curvature point of view, conformal flatness is equivalent to the vanishing of the Weyl and the Cotton tensor. In particular, the Riemann tensor can be recovered by its trace part, namely the Ricci tensor. Schoen and Yau [26] showed that conformal flatness together with (constant) positive scalar curvature still allows much flexibility. In contrast, conditions on the Ricci curvature put strong restrictions on the geometry of the manifold. Tani [27] proved that any compact conformally flat n-dimensional manifold with positive Ricci curvature and constant positive scalar curvature is covered isometrically by Sn with the round metric.
    [Show full text]
  • Deformations with Maximal Supersymmetries Part 2: Off-Shell Formulation
    Deformations with maximal supersymmetries part 2: off-shell formulation The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Chang, Chi-Ming et al. “Deformations with Maximal Supersymmetries Part 2: Off-Shell Formulation.” Journal of High Energy Physics 2016.4 (2016): n. pag. As Published http://dx.doi.org/10.1007/JHEP04(2016)171 Publisher Springer Berlin Heidelberg Version Final published version Citable link http://hdl.handle.net/1721.1/103304 Terms of Use Creative Commons Attribution Detailed Terms http://creativecommons.org/licenses/by/4.0/ Published for SISSA by Springer Received: February 26, 2016 Accepted: April 23, 2016 Published: April 28, 2016 Deformations with maximal supersymmetries part 2: JHEP04(2016)171 off-shell formulation Chi-Ming Chang,a Ying-Hsuan Lin,a Yifan Wangb and Xi Yina aJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. bCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Continuing our exploration of maximally supersymmetric gauge theories (MSYM) deformed by higher dimensional operators, in this paper we consider an off- shell approach based on pure spinor superspace and focus on constructing supersymmetric deformations beyond the first order. In particular, we give a construction of the Batalin- Vilkovisky action of an all-order non-Abelian Born-Infeld deformation of MSYM in the non-minimal pure spinor formalism. We also discuss subtleties in the integration over the pure spinor superspace and the relevance of Berkovits-Nekrasov regularization.
    [Show full text]
  • Hypersurfaces of Einstein Manifolds
    ANNALES SCIENTIFIQUES DE L’É.N.S. NORIHITO KOISO Hypersurfaces of Einstein manifolds Annales scientifiques de l’É.N.S. 4e série, tome 14, no 4 (1981), p. 433-443 <http://www.numdam.org/item?id=ASENS_1981_4_14_4_433_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1981, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scient. EC. Norm. Sup., 40 serie, t. 14, 1981, p. 433 a 443. HYPERSURFACES OF EINSTEIN MANIFOLDS BYNORIHITO KOISO(1) 0. Introduction and results Let (M, g) be an Einstein manifold of dimension n +1 (n ^ 2). We consider certain classes of hypersurfaces in (M, g). First, let (M, g} be a totally umbilical hypersurface in (M, g), i.e., we assume that the second fundamental form a satisfies ^=jg for some function y on M. If we know completely the curvature tensor of (M, g}, we can get much information on (M, g). For example, if(M, g) is a symmetric space, then (M, g) is also a locally symmetric space, and so the classification of such pairs [(M, g), (M, g)} reduces to Lie group theory (see Chen [4] (2), Chen and Nagano [5], Naitoh [10]).
    [Show full text]
  • Hep-Th/0505211V3 12 Dec 2005 Sebasti´An Franco Brane and Tilings Geometry Toric from Theories Gauge VERSION HYPER - Style
    Preprint typeset in JHEP style. - HYPER VERSION MIT-CTP-3646, CERN-PH-TH/2005-084, HUTP-05/A0027 Gauge Theories from Toric Geometry and Brane Tilings Sebasti´an Franco1, Amihay Hanany1, Dario Martelli2, James Sparks3, David Vegh1, and Brian Wecht1 1. Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. 2. Department of Physics, CERN Theory Division, 1211 Geneva 23, Switzerland. 3. Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02318, U.S.A. and Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] arXiv:hep-th/0505211v3 12 Dec 2005 Abstract: We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi–Yau singularity. Our method combines information from the geometry and topology of Sasaki–Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3–branes probing a toric Calabi–Yau singularity can be deduced. The infinite family of toric singularities with known horizon Sasaki–Einstein manifolds La,b,c is used to illustrate these ideas. We construct the corresponding quiver gauge theories, which may be fully specified by giving a tiling of the plane by hexagons with certain gluing rules. As checks of this construction, we perform a-maximisation as well as Z-minimisation to compute the exact R-charges of an arbitrary such quiver. We also examine a number of examples in detail, including the infinite subfamily La,b,a, whose smallest member is the Suspended Pinch Point.
    [Show full text]
  • Oxford Physics Department Notes on General Relativity Steven Balbus
    Oxford Physics Department Notes on General Relativity Steven Balbus Version: January 25, 2017 1 These notes are intended for classroom use. There is little here that is original, but a few results are. If you would like to quote from these notes and are unable to find an original literature reference, please contact me by email first. Thank you in advance for your cooperation. Steven Balbus 2 Recommended Texts Hobson, M. P., Efstathiou, G., and Lasenby, A. N. 2006, General Relativity: An Introduction for Physicists, (Cambridge: Cambridge University Press) Referenced as HEL06. A very clear, very well-blended book, admirably covering the mathematics, physics, and astrophysics of GR. Excellent presentation of black holes and gravitational radiation. The explanation of the geodesic equation and the affine connection is very clear and enlightening. Not so much on cosmology, though a nice introduction to the physics of inflation. Overall, my favourite text on this topic. (The metric has a different sign convention in HEL06 compared with Weinberg 1972 & MTW [see below], as well as these notes. Be careful.) Weinberg, S. 1972, Gravitation and Cosmology. Principles and Applications of the General Theory of Relativity, (New York: John Wiley) Referenced as W72. What is now the classic reference by the great man, but lacking any discussion whatsoever of black holes, and almost nothing on the geometrical interpretation of the equations. The author is explicit in his aversion to anything geometrical: gravity is a field theory with a mere geometrical \analogy" according to Weinberg. But there is no way to make sense of the equations, in any profound sense, without immersing onself in geometry.
    [Show full text]